Abstract

We study a Hamiltonian describing a pendulum coupled with several anisochronous oscillators, devising an expansion for the splitting matrix associated with a homoclinic point. This expansion consists of contributions that are manifestly exponentially small in the limit of vanishing hyperbolicity by a shift-of-contour argument. An exponentially small upper bound on the splitting is implied. The focus of this paper is on the method.

I am indebted to Antti Kupiainen for all his advice. I express my gratitude to Guido Gentile, Kari Astala, and Jean Bricmont for their comments. I thank Emiliano De Simone and Giovanni Gallavotti for useful discussions. Mischa Rudnev, Pierre Lochak, and Michela Procesi were kind enough to explain to me some of their works on the subject. I am grateful to Joel Lebowitz and Rutgers University as well as Lai-Sang Young and the Courant Institute for their hospitality during the final stages of writing this work. Partial support was received from the Finnish Cultural Foundation, the Finnish Academy of Science and Letters, the Academy of Finland, and NSF under Grant No. DMR-01-279-26.

Article outline:I. MAIN CONCEPTS AND RESULTSA. BackgroundB. Guideline to the paperC. The modelD. Description of resultsII. SIZE OF THE HOMOCLINIC SPLITTINGA. The Melnikov termB. Regularized integralsC. Asymptotic expansion for the splitting matrixD. Emergence of exponential smallnessIII. PROOF OF THEOREM 4A. Analytic structure of the solutionB. Tree expansionC. Some combinatoricsD. Simplification of integrals: Scalar treesE. Integration by parts: One stepF. Integration by parts: Exhausting the entire treeG. EstimatesH. Remaining integralsIV. DISCUSSION